TheoremHartogs (1906): any holomorphic functions on are analytically continued to . Namely, there is a holomorphic function on such that on .

In fact, using the Cauchy integral formula we obtain the extended function , All holomorphic functions are analytically continued to the polydisk, which is restrictly larger than the domain on which the original holomorphic function is defined. Such phenomenon never happen in the case of one variable.

Let f be a holomorphic function on a setG\K, where G is an open subset of Cn (n ≥ 2) and K is a compact subset of G. If the relative complementG\K is connected, then f can be extended to a unique holomorphic function on G.

The theorem does not hold when n = 1. To see this, it suffices to consider the function f(z) = z−1, which is clearly holomorphic in C\{0}, but cannot be continued as an holomorphic function on the whole C. Therefore the Hartogs' phenomenon constitutes one elementary phenomenon that emphasizes the difference between the theory of functions of one and several complex variables.

^See for example Vladimirov (1966, p. 153), which refers the reader to the book of Fuks (1963, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).